I think the founders established the council, and the surviving 12 elect to appoint a new one every time one dies or retires. You probably don't retire from this job. I don't like the part of Ecce Perago where they all die, but it is pretty influential to my headcanon, except that I firmly believe that there are 13 of them…

In my head, they are picked from The lowest of the low, People who have nothing to loose or even owners of companies. I think that they are there just to look mysterious, being that O5-13 the Persian is a double mystery and O5-11, who I call Roger Sheldon as examples, are just normal people who were selected to give speeches for whoever is truly pulling the strings. I also believe the administrator is one of them. I think that someone as high as an O5 wouldn’t be documented for security purposes, and they may not even be that high of a level as some think. I believe that the O5 are basically puppets.

In my headcanon they are chosen from the top echelon of humanity, whether it be the greatest scientist (a matter of semantics) or the brightest of world leaders. The O5 are the people that maintain the illusion of normalcy and as such must be able to decide what is best for humanity in it's entirety, no detail too small and no cost too severe.

Given that O5s aren't supposed to ever have contact w/ SCPs, I don't think they're promoted from within, as anyone who would be worthy has already had contact w/ SCPs and could theoretically be contaminated by unknown or long-term effects (see SCP-884 or ANY infohazard). I imagine that each O5 has a list of possible successors, made shortly after s/he becomes an O5, and when s/he retires or dies, the rest of the council deliberates on which one to invite/recruit to become a member of the council.

That being said, there are almost certainly bugs in the system such that at least ONE empowered/affected individual has made it onto the council.

Recruited from anywhere, as long as they've proven their management skills and ability to be an uncaring bastard when required.

So, basically the same as anyone else in the Foundation. For the recruitment myself, I'm seeing a bit of a Redpill, Bluepill style Matrix thing going on, with some Class A Amnesiacs mixed in here and there.

New idea: one of the O5s is a perfectly ordinary 7 year old child. All containment procedures and operations plans are run by them before institution, so as to weed out the mind-numbingly stupid ones first.

I like to imagine that the O5s are from all different ethnic backgrounds, genders, ages, and economic status so as to eliminate bias. On the one hand, you might have a scientific genius with tons of money. On the other, an ordinary student at a community college somewhere. So while some of the O5s are remarkable people with special skill sets, others are not particularly special, but help balance things out in terms of morals. The perspective they're trying to get is not just "What would a group of some of the top members of society do?" but "What would a group of human beings do?" Let's say Scientific Genius up there wants to continue experimentation that is harmful to the subject, but For The Greater Good. Ordinary College Student could step in and point out the ethical ramifications of such a thing. It's a really weird system of checks and balances.

The O5 council is, in my head composed of experts on technical, military and scientific matters, some of them advancing from former high-level researchers and some of them maybe taken from outside.
They are not about perspective, or any other such bullshit, but about ensuring something resembling normality is maintained as far as possible and that the world doesn't go down the crapper. Such a thing demands quick, rational decisionmaking that random members of the public simply lack. It's precisely why there aren't O5 council elections every five years or so.
You wouldn't want members of the ordinary public to eg. direct the operation of a nuclear power plant. Why involve them in something that's so much more dangerous?